By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple:
Is there a logic $\mathcal{L}$ which is fully compact and such that no properly-stronger logic is fully compact?
It seems like there should be an easy negative answer as long as the logics are required to be set-sized, but I can't quite get the details to work; the relativizability property of regular logics makes everything rather tedious. And if we allow class-sized logics (e.g. $\mathcal{L}_{\infty,\omega}$, although of course that's not fully compact) then this idea goes out the window and I have no intuition.
I'm happy to add any definability criteria which would help, but I do not want to add Lowenheim-Skolem-type conditions. In particular, the results of Shelah/Vaananen (A note on extensions of infinitary logic) don't seem to apply here.
